zbMATH — the first resource for mathematics

Asymptotics for large time of solutions to nonlinear system associated with the penetration of a magnetic field into a substance. (English) Zbl 1224.35189
Summary: The nonlinear integro-differential system associated with the penetration of a magnetic field into a substance is considered. The asymptotic behavior as \(t\to \infty \) of solutions for two initial-boundary value problems are studied. The problem with non-zero conditions on one side of the lateral boundary is discussed. The problem with homogeneous boundary conditions is also studied. The rates of convergence are given. Results presented show the difference between stabilization characters of solutions of these two cases.

35K55 Nonlinear parabolic equations
45K05 Integro-partial differential equations
74H40 Long-time behavior of solutions for dynamical problems in solid mechanics
78A30 Electro- and magnetostatics
Full Text: DOI EuDML
[1] A. L. Amadori, K.H. Karlsen, C. La Chioma: Non-linear degenerate integro-partial differential evolution equations related to geometric Lévy processes and applications to backward stochastic differential equations. Stochastics Stochastics Rep. 76 (2004), 147–177. · Zbl 1049.60050 · doi:10.1080/10451120410001696289
[2] J.M. Chadam, H.M. Yin: An iteration procedure for a class of integrodifferential equations of parabolic type. J. Integral Equations Appl. 2 (1990), 31–47. · Zbl 0701.45004 · doi:10.1216/JIE-1989-2-1-31
[3] B.D. Coleman, M. E. Gurtin: On the stability against shear waves of steady flows of non-linear viscoelastic fluids. J. Fluid Mech. 33 (1968), 165–181. · Zbl 0207.25302 · doi:10.1017/S0022112068002430
[4] C.M. Dafermos: An abstract Volterra equation with application to linear viscoelasticity. J. Differ. Equations 7 (1970), 554–569. · Zbl 0212.45302 · doi:10.1016/0022-0396(70)90101-4
[5] C. Dafermos: Stabilizing effects of dissipation. Proc. Int. Conf. Equadiff 82, Würzburg 1982. Lect. Notes Math. Vol. 1017. 1983, pp. 140–147.
[6] C.M. Dafermos, J.A. Nohel: A nonlinear hyperbolic Volterra equation in viscoelasticity. Contributions to analysis and geometry. Suppl. Am. J. Math. (1981), 87–116.
[7] H. Engler: Global smooth solutions for a class of parabolic integrodifferential equations. Trans. Am. Math. Soc. 348 (1996), 267–290. · Zbl 0848.45002 · doi:10.1090/S0002-9947-96-01472-9
[8] H. Engler: On some parabolic integro-differential equations: Existence and asymptotics of solutions. Proc. Int. Conf. Equadiff 82, Würzburg 1982. Lect. Notes Math. Vol. 1017. 1983, pp. 161–167.
[9] D.G. Gordeziani, T.A. Jangveladze (Dzhangveladze), T.K. Korshiya: Existence and uniqueness of the solution of certain nonlinear parabolic problems. Differ. Equations 19 (1983), 887–895. · Zbl 0582.35065
[10] G. Gripenberg: Global existence of solutions of Volterra integrodifferential equations of parabolic type. J. Differ. Equations 102 (1993), 382–390. · Zbl 0780.45012 · doi:10.1006/jdeq.1993.1035
[11] G. Gripenberg, S.-O. Londen, O. Staffans: Volterra Integral and Functional Equations. Encyclopedia of Mathematics and Its Applications, Vol. 34. Cambridge University Press, Cambridge, 1990. · Zbl 0695.45002
[12] M.E. Gurtin, A.C. Pipkin: A general theory of heat conduction with finite wave speeds. Arch. Ration. Mech. Anal. 31 (1968), 113–126. · Zbl 0164.12901 · doi:10.1007/BF00281373
[13] T.A. Jangvelazde (Dzhangveladze): On the solvability of the first boundary value problem for a nonlinear integro-differential equation of parabolic type. Soobsch. Akad. Nauk Gruz. SSR 114 (1984), 261–264. (In Russian.)
[14] T.A. Jangveladze (Dzhangveladze), Z.V. Kiguradze: Asymptotic behavior of the solution of a nonlinear integro-differential diffusion equation. Differ. Equ. 44 (2008), 538–550. · Zbl 1172.35500 · doi:10.1134/S0012266108040083
[15] T.A. Jangveladze (Dzhangveladze), Z.V. Kiguradze: Asymptotics of a solution of a nonlinear system of diffusion of a magnetic field into a substance. Sib. Mat. Zh. 47 (2006), 1058–1070 (In Russian.);, English translation: Sib. Math. J. 47 (2006), 867–878. · Zbl 1150.45316
[16] T.A. Jangveladze (Dzhangveladze), Z.V. Kiguradze: Estimates of the stabilization rate as t of solutions of the nonlinear integro-differential diffusion system. Appl. Math. Inform. Mech. 8 (2003), 1–19. · Zbl 1070.45009
[17] T.A. Jangveladze (Dzhangvelazde), Z.V. Kiguradze: On the stabilization of solutions of an initial-boundary value problem for a nonlinear integro-differential equation. Differ. Equ. 43 (2007), 854–861;, Translation from Differ. Uravn. 43 (2007), 833–840. (In Russian.) · Zbl 1136.78018 · doi:10.1134/S0012266107060110
[18] T.A. Jangveladze (Dzhangvelazde), B.Ya. Lyubimov, T.K. Korshiya: Numerical solution of a class of non-isothermal diffusion problems of an electromagnetic field. Tr. Inst. Prikl. Mat. Im. I.N. Vekua 18 (1986), 5–47. (In Russian.)
[19] J. Kačur: Application of Rothe’s method to evolution integrodifferential equations. J. Reine Angew. Math. 388 (1988), 73–105. · Zbl 0638.65098
[20] L.D. Landau, E.M. Lifshitz: Electrodynamics of Continuous Media. Pergamon Press, Oxford-London-New York, 1960. · Zbl 0122.45002
[21] G. Laptev: Mathematical singularities of a problem on the penetration of a magnetic field into a substance. Zh. Vychisl. Mat. Mat. Fiz. 28 (1988), 1332–1345 (In Russian.); English translation:, U.S.S.R. Comput. Math. Math. Phys. 28 (1990), 35–45.
[22] G. Laptev: Quasilinear parabolic equations which contains in coefficients Volterra’s operator. Math. Sbornik 136 (1988), 530–545 (In Russian.);, English translation: Sbornik Math. 64 (1989), 527–542.
[23] J.-L. Lions: Quelques méthodes de résolution des problèmes aux limites non-linéaires. Dunod/Gauthier-Villars, Paris, 1969. (In French.)
[24] N.T. Long, A.P.N. Dinh: Nonlinear parabolic problem associated with the penetration of a magnetic field into a substance. Math. Methods Appl. Sci. 16 (1993), 281–295. · Zbl 0797.35099 · doi:10.1002/mma.1670160404
[25] N.T. Long, A.P.N. Dinh: Periodic solutions of a nonlinear parabolic equation associated with the penetration of a magnetic field into a substance. Comput. Math. Appl. 30 (1995), 63–78. · Zbl 0834.35070 · doi:10.1016/0898-1221(95)00068-A
[26] R.C. MacCamy: An integro-differential equation with application in heat flow. Q. Appl. Math. 35 (1977), 1–19.
[27] M. Renardy, W. J. Hrusa, J.A. Nohel: Mathematical Problems in Viscoelasticity. Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 35. Longman Scientific & Technical/John Wiley & Sons, Harlow/New York, 1987. · Zbl 0719.73013
[28] M. Vishik: Über die Lösbarkeit von Randwertaufgaben für quasilineare parabolische Gleichungen höherer Ordnung (On solvability of the boundary value problems for higher order quasilinear parabolic equations). Mat. Sb. N. Ser. 59 (1962), 289–325. (In Russian.) · Zbl 0149.31202
[29] H.M. Yin: The classical solutions for nonlinear parabolic integrodifferential equations. J. Integral Equations Appl. 1 (1988), 249–263. · Zbl 0671.45004 · doi:10.1216/JIE-1988-1-2-249
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.